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6.4.2 Roots of Complex Numbers
Given the value of the complex number z, we are interested here in finding
the solutions of the equation:

n
v = z (6.28)
Let us write both the solutions and z in polar forms,

v = ρ(cos( α) + j sin( α)) (6.29)

z = r(cos( )θ + j sin( ))θ (6.30)

n
From the De Moivre theorem, the expression for v = z can be written as:
n
ρ (cos( n α) + j sin( n α)) = r(cos( θ) + j sin( θ)) (6.31)

Comparing the moduli of both sides, we deduce by inspection that:

n
ρ= r (6.32)

The treatment of the argument should be done with great care. Recalling
that two angles have the same cosine and sine if they are equal or differ from
each other by an integer multiple of 2π, we can then deduce that:

θ
nα =+ 2 kπ k = 0, ± 1, ± 2, ± 3, … (6.33)
Therefore, the general expression for the roots is:

n   θ k 2 π   θ k 2 π 
z 1/ n = r 1/   cos  n + n  + j sin  n + n   (6.34)

n 1)
with k = 01 2,, ,… ,( −
Note that the roots reproduce themselves outside the range: k = 0, 1, 2, …,
(n – 1).

In-Class Exercises

5
Pb. 6.19 Calculate the roots of the equation z – 32 = 0, and plot them in the
complex plane.

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